Let's assume that p+q+r is odd. We will show that an odd number of p,q,r must be odd by showing that it can not be the case that 0 or 2 of them are odd.
If none of them are odd, then they are all even and the sum of three evens is even, contradicting the fact that p+q+r is odd.
If two of them are odd, let's say p and q are both odd. Then r is even and p+q is even and so p+q+r = even, contradicting the fact that p+q+r must be odd.
Therefore it must be the case that the number of odd numbers among p,q,r must be either 1 or 3.